Abstract
Lattice vibration modes are collective excitations in periodic arrays of atoms or molecules. These modes determine novel transport properties in solid crystals. Analogously, in periodical arrangements of magnetic vortexstate disks, collective vortex motions have been predicted. Here, we experimentally observe wave modes of collective vortex gyration in onedimensional (1D) periodic arrays of magnetic disks using timeresolved scanning transmission xray microscopy. The observed modes are interpreted based on micromagnetic simulation and numerical calculation of coupled Thiele equations. Dispersion of the modes is found to be strongly affected by both vortex polarization and chirality ordering, as revealed by the explicit analytical form of 1D infinite arrays. A thorough understanding thereof is fundamental both for lattice vibrations and vortex dynamics, which we demonstrate for 1D magnonic crystals. Such magnetic disk arrays with vortexstate ordering, referred to as magnetic metastructure, offer potential implementation into information processing devices.
Introduction
Recently, collective spin excitations in nanoscale magnetic elements, particularly spin waves, have become a focus of attention in nanomagnetism and related spintronics, owing to their potential implementation in information processing devices^{1,2,3,4,5,6,7,8,9,10,11,12}, in addition to the advances made in the understanding of fundamental modes in such geometrically confined spin systems^{13}. For example, some earlier works^{14,15} propose magnetic quantum dot cellular automata that consist of single domain magnets for alternative informationsignal propagation. New advances in both nanofabrication technology^{16} and time and spaceresolved measurement techniques^{1,2,17} have enabled intensive studies of a wide variety of magnonic crystals (MCs) such as onedimensional (1D) strips^{18,19,20,21,22}, twodimensional (2D) arrays of magnetic nanoelements^{23,24,25,26} and antidot lattices of periodic holes having a circular or rectangular shape in 2D continuous films^{27,28,29}. Furthermore, technological interest in the practical applicability of MCs to future information storage and processing devices^{30,31} is rapidly growing. In patterned MCs, band structures including band widths and gaps can, in principle, be tailored through their constituent materials and the isolated elements' dimension and separation distances^{1,2,3,18,19,20,21,22,23,24,25,26,27}. However, despite recent insight into the allowed magnonic modes in a rich variety of MCs, collective vortexgyration modes in vortexstate arrays remain elusive, notwithstanding Shibata et al.^{32,33}′s theoretical prediction of dipolarcoupled vortices in 2D magnetic disk arrays and the experimental demonstrations of vortexgyration transfer between two (or more) coupled disks^{34,35,36,37,38,39,40,41,42,43,44,45}.
Here, we report on the first direct experimental demonstration, by means of stateoftheart timeresolved scanning transmission xray microscopy (STXM), of quantized (or discrete) wave modes of collective vortex gyrations excited in five physically separated but dipolarcoupled disks in a permalloy (Py: Ni_{80}Fe_{20}) disk array. With the help of numerical calculations of coupled linearized Thiele equation, micromagnetic numerical simulations and analytical derivations, we investigate the experimentally observed discrete modes and their dispersion relations. The results reveal that characteristic dispersions are expressed in terms of the intrinsic angular eigenfrequency ω_{0} of isolated disks and the specific polarization p and chirality C ordering. The underlying physics can be well understood in terms of the dynamic dipolar interaction associated with the specific p and C orderings. Accordingly and promisingly, the propagation property of collective vortex gyration and its dispersion can be manipulated by vortexstate ordering, the dimensions of each disk and the nearestneighbouring (NN) disks's interdistance. This work constitutes a milestone towards the practical achievement of this new class of MCs harnessing their advantages.
Results
Sample structure and STXM measurements
Figure 1 shows a scanning electron microscopy (SEM) image of the sample (Fig. 1a) as well as STXM images of outofplane core magnetizations (Fig. 1b) and inplane curling magnetizations (Fig. 1c) in each of the five Py disks (see Methods for the sample dimensions). Here, the polarization and chirality configurations of the array are p = [+1,−1, +1, −1, +1] and C = [−1, −1, −1, −1, +1], respectively (see Fig. 1d), as obtained from the STXM images, where p = +1(−1) corresponds to upward (downward) core magnetization and C = +1 to counterclockwise (CCW) and C = −1 to clockwise (CW) inplane curling magnetization. Note that the sample has the opposite core orientations between the NN disks.
In order to trigger an excitation of vortex gyration in the first disk, we launch a current pulse of 1.8 ns duration into the electrode stripline, resulting in a field pulse of about 2.4 mT strength [see the corresponding inset of Fig. 1(a)]. The propagation of vortex gyration excited at the first disk is driven by dipolar interaction between the NN disks where individual cores are shifted from their static center positions, thereby yielding a nonzero effective inplane magnetization. Oscillatory motions of the individual cores are measured by STXM operated in the pumpandprobe sampling mode, which allows for imaging of the cores' outofplane magnetizations utilizing elementspecific Xray magnetic circular dichroism (XMCD) as magnetic contrast at a lateral resolution of about 25 nm and a temporal resolution as low as 35 ps (for further details, see Methods).
Vortexcore gyration propagation along dipolarcoupled disks
Figure 2 shows the x (red color) and y (blue) components (Fig. 2a) of the displacements of the individual cores and their trajectories (Fig. 2b) in the disk plane, as measured by timeresolved STXM (see also Supplementary Movie 1). The experimental results (top of 2a and 2b) are compared with the corresponding micromagnetic simulations (bottom of 2a and 2b) performed using the OOMMF code (version 1.2a4)^{46}. The characteristic beating patterns along with their modulation envelopes are observed in each of the five disks (Fig. 2a). Owing to the direct excitation of the first disk, a largeamplitude gyration in that disk is observed and is then allowed to propagate towards the NN disk and beyond through the array. The vortexgyration transfer to the next disk and its further propagation are evidenced by the increase of the gyration amplitude in the second and remaining disks along with the concomitant and remarkable decrease of the first disk's gyration amplitude. The ratio between the maximum displacements in disk 5 and disk 1 is about 0.24. Since our pumpandprobe measurements are carried out within a time period of 60.8 ns and the intrinsic damping of Py is not negligible but rather significant (as strong as α ~ 0.01), we cannot clearly observe backward propagation bounced at the last (5^{th}) disk. However, the signature of weak reflection is evident by the increase of the gyration amplitude in the 4^{th} disk at around 55 ns, as compared with the simulation result.
It has been reported that coupled gyrations in twodipolarcoupled disks can be described by the superposition of the two normal modes^{32,35,36,38,39}. Dipolar interaction between NN disks breaks the radial symmetry of the potential energy of each core, which depends on the disk pair's relative vortexstate configuration (both the polarization and chirality ordering). Analogously, for the case of the fivedisk system used in this study, the beating patterns is the result of linear combinations of the five normal modes of coupled vortex gyration in the entire array [for more information, see Supplementary Information B].
Discrete wave modes of collective vortex gyration
To illustrate the collective vortexgyration modes excited in the real sample, in Fig. 3(a) we plot the frequency spectra (red circles) of core motions in the individual disks as obtained from fast Fourier transformation (FFT) of the core position vector X_{n} multiplied by C_{n} in the n^{th} disk. We also compare the experimental results with micromagnetic simulation (green circles) for the fivedisk model system and numerical calculation (blue circles) based on five coupled linearized Thiele equations^{47} (see Supplementary Information A). Because of the intrinsic damping of core gyration in isolated disks, the peaks are broadened and overlapped with neighboring peaks to an extent, that they cannot be separated. Further deviations between the experimental data and micromagnetic simulations as well as numerical calculations can be attributed to sample imperfections and the chosen time steps (400 ps) between the snapshot images taken by STXM. Specifically, with regard to disks 2 and 4, only one peak of wide width appears in the experimental data whereas two peaks appear in the simulations and numerical data. In contrast, two clear peaks and a very weak third peak appear in disks 1 and 3, which is in quantitative agreement with the micromagnetic simulation and numerical calculation.
In order to clarify the presence of fundamental discrete modes, we conduct further numerical calculation of coupled Thiele equation on a fivedisk model of the same dimensions and material parameters as those in the real sample, but with zero damping. The right panel of Fig. 3a shows the characteristic frequency spectrum of each disk. From disk 1 through disk 5, different peaks of contrasting FFT powers are observed. All of the five distinct peaks marked by ω_{i} (where i = 1, 2, 3, 4, 5) are shown in disks 1 and 5. By contrast, the ω_{3} peak disappears in disks 2 and 4, while the ω_{2} and ω_{5} peaks disappear in disk 3. Each of the peaks of all of the modes is located at the same position in all of the disks.
From the inverse FFT of all of the peaks of each mode, we can extract the spatial correlations of core motions in the individual disks for each mode ω_{i}. Figure 3(b) shows the trajectories of the orbiting cores in motion in the individual disks along with the profiles of the C_{n}Y_{n} component of the core positions in the fivedisk array. For all of the modes ω_{i}, the individual core' gyration amplitudes are markedly distinct among the disks and modes. More interestingly, the collective motions of the individual cores in the whole array represent certain wave forms of different wavelengths. The gyration amplitudes for all of the modes are symmetric with respect to the center of the array and are also completely pinned at imaginary disks at both ends, denoted disk 0 and disk 6 for the case of N = 5. These features represent a standingwave form of a certain wavelength in terms of collective vortexgyration motions, being quite analogous to a string, the ends of which are attached to the left and right walls respectively, thus having no displacement. Accordingly, we can interpret the collective and discrete wave modes as in the nodamping case, based on the fixed boundary condition in such a 1D array of finite disk number N. In this case, the boundary condition is given as , where ψ denotes the displacement of beads, N is the number of elements and d_{int} is the interdistance between the elements. From this boundary condition, the wave vectors of the allowed modes can be expressed simply as , where m = 1, 2, … N1 and N (For more information, see Supplementary Information B). Thus, the discrete (quantized) five modes' wave numbers of the collective vortex gyrations in the fivedisk array are coincident with the values of , where m = 1, 2, 3, 4, 5.
Dispersion relation in coupled fivedisk array
As described above, collective vortexgyration modes represent standing waves of discrete wavelengths (i.e., quantized k values). Here, to extract the dispersion (ω  k relation) of all of the modes, we perform FFTs of the collective core profiles for the individual modes according to (where m = 1, 2, 3, 4, 5) with a fixed value of d_{int} = 2250 nm for the real sample. Figure 4 shows the FFT powers in the ωk spectra obtained from the experimental data, micromagnetic simulation (for α = 0.01) and numerical calculation of coupled Thiele equation (for both cases of α = 0.01 and α = 0). FFTs of each of the X_{n} and Y_{n}, multiplied by C_{n} (i.e., C_{n}X_{n} and C_{n}Y_{n}), are performed. Using such reduced parameters of C_{n}X_{n} and C_{n}Y_{n}, we can consider only the polarity ordering for comparison between experimental and numerical calculation data (see Supplementary Information A).
The overall shape of the dispersion from the experimental data qualitatively agrees well with those from the micromagnetic simulations and numerical calculation, though they show quantitative discrepancy in the frequency and FFT power between each mode. As already mentioned above, discrepancies might be associated with the chosen measurement parameters, sample imperfections as well as a difference in the saturation magnetization between the experiment and micromagnetic simulations. The white solid lines indicate the result of an analytically derived explicit form for a 1D infinite array (for the calculation, see Supplementary Information C). As noted earlier, the intrinsic damping of vortexcore gyration in isolated disks causes the broadening of the ω values (see Fig. 4). For the case of nodamping, five discrete quantized modes without the ω–value broadening are distinctly shown in the spectra (right panel).
Next, note that the overall shape of dispersion is concave down, that is, a higher frequency at k = 0 and a lower frequency at k = π/d_{int} for the case of C_{n}X_{n} and concave up (vice versa) for C_{n}Y_{n}. This reversal between C_{n}X_{n} and C_{n}Y_{n} can be understood in terms of the latticenumberdependent phase difference between the x and y components of the vortexcore positions. Since the gyration's rotational sense is determined by the polarization p of a given disk, the phase difference between the x and y components of the core position vector in the n^{th} disk is given as the product of and . Accordingly, for the case of the antiparallel polarization between the NN disks as in the sample, the phase difference between the x and y components can be expressed as , where . This results in the shift of the kvector in reciprocal space, as . Considering the real value of = 1.3963 μm^{−1} for d_{int} = 2250 nm), the experimental data are fully consistent for the k shift by between C_{n}X_{n} and C_{n}Y_{n}, as shown in Fig. 4.
Extension to semiinfinite or infinite 1D magnonic crystals
Based on the above approach, we can extend to a array system comprised of a semiinfinite or infinite 1D array composed of periodically arranged disks (referred to as 1D MCs). Specifically, we accomplish this by numerical calculation of a large number of disks (here, N = 201) and an analytically derived dispersion equation for infinite arrays. Here we also consider specific parallel and antiparallel ordering of both the p and C configurations between the NN disks: Type I: [p_{n}, C_{n}] = [(−1)^{n}^{+1},1] for the antiparallel p and parallel C ordering; Type II: [(−1)^{n+1}, (−1)^{n+1}] for the antiparallel p and C ordering; Type III: [1,1] for the parallel p and C ordering and Type IV: [1, (−1)^{n+1}] for the parallel p and antiparallel C ordering. Considering those additional degrees of freedom for both the p and C ordering, we analytically derive an explicit dispersion relation based on linearized Thiele equations of coupled vortexcore motions, taking into account the potential energy modified by dipolar interaction between only NN disks^{32,33}. Here, for simplicity, we assume 1D arrays of an infinite number of equaldimension disks. For zero damping (α = 0), the dispersion relation can be written as with and , where is the stiffness coefficient of the potential energy for isolated disks. and represent the interaction strength along the x (here x is the bonding axis) and y axes, respectively (for the detailed derivation procedure, see Supplementary Information C). p_{n}p_{n}_{+1} = +1(−1) and C_{n}C_{n+}_{1} = 1(−1) indicate parallel (antiparallel) p and C ordering, respectively, between the NN disks. In this case, the wave vector k has a continuous value due to the infinite number of existing modes in such an infinite 1D array. This explicit analytical form indicates that the dispersion relation is a function of an isolated disk's eigenfrequency ω_{0} and the coupling strength between the NN disks, that is, and , as well as those special p and C ordering.
The numerical calculation of the analytical form of for four different types of vortexstate ordering noted above are displayed by the white lines in Fig. 5a, which are in excellent agreement with the dispersion spectrum from the FFTs of the X_{n} components of the individual disks, which are obtained from the numerical calculation of N coupled Thiele equations for the N = 201 system with damping (α = 0.01). While performing the FFTs, we imposed a periodic boundary condition to describe such a semiinfinite system in terms of traveling waves. Accordingly, the resultant kvalues are given as , where m is any integer value under the constraint of . All of the dispersion curves are symmetric with respect to k = 0, because the gyration is supposed to propagate from the center towards both ends.
We stress here that the overall shape of dispersion is determined by the sign of p_{n}p_{n+1}C_{n}C_{n+1}; such that concave up for p_{n}p_{n+1}C_{n}C_{n+1} = 1 and concave down for p_{n}p_{n+1}C_{n}C_{n+1} = −1. Also, the band width is wider for the case of the antiparallel p ordering than for the parallel p ordering. The bandwidth variation in the p ordering reflects the fact that the opposite polarization between NN disks has a stronger dipolar interaction (resulting in large frequency splitting) than does the same polarization, as noted in earlier reports^{35,36,37,38}. This is caused by the rotational sense of stray fields around a given disk that is opposite to that of the core gyration. A rotating field efficiently couples into the circular eigenmode of gyration when the sense of rotation of the stray field coincides with the sense of gyration^{46}. Consequently, the dipolar interaction between NN disks having the opposite polarization is stronger than that between those having the same polarization.
For a comprehensive understanding of dispersion variation according to the sign of p_{n}p_{n+1}C_{n}C_{n+1}, we extract the spatial distributions of collective vortexcore motions from the analytical derivation at specific values of k = 0 (red lines) and k = k_{BZ =} π/d_{int} (blue lines), as shown in Fig. 5(b). We calculate the dynamic dipolar interaction energy densities as a function of time for four different types of vortexstate ordering. The insets show the effective inplane magnetizations of a given disk and both NN disks around it in a unit time period of 2π/ω. The rotating effective magnetizations <M_{n}> (graycoloured wide arrows in each disk) in the NN disks and their relative orientations determine the characteristic dispersion that varies with both p and C orderings. The governing rule is determined by the dynamic dipolar interaction energy term: for the case where the NN disks' <M_{n}> is in parallel (antiparallel) orientation along the x axis, their dipolar interaction energy is at the lowest (highest) energy level, whereas for the case where their relative <M_{n}> is in parallel (antiparallel) orientation along the y axis, the energy is at the second highest (second lowest) energy level, as represented by the gray arrows in the three disks in Fig. 5b. Thus, the phase relation of NN disk's <M_{n}> is crucial to dynamic dipolar interaction, the overall value of which during a unit period is determined by C_{n}C_{n+1} as well as p_{n}p_{n+1} (For more quantitative interpretation, see Supplementary Information D).
Discussion
We experimentally observed the wave modes of collective vortexcore gyration excitation along with their quantization and their dispersions in a array of five coupled disks. With the help of the analytical derivation, numerical calculation of coupled Thiele equation and micromagnetic simulation, those discrete modes can be well understood in terms of the relative orientations of rotating effective inplane magnetizations and the dynamic dipolar interaction between the individual disks. Additional degrees of freedom of vortexstate ordering, including polarization and chirality, dramatically affect the phase relation of the dynamic dipolar interaction, thereby leading to changes in dispersion. Analogous to quantized latticevibration modes in solid crystals, the wave modes of vortex gyrations in periodically patterned magnetic dots are fundamental. This work enables the extension of coupled vortex disks to new types of MCs composed of ordered vortexstate disks, thus opening the way to control vortexgyration propagations, band gaps and widths of dispersions in 1D or 2D MCs.
Such newtype MCs might offer the advantages of limitless switchablevortexstate and vortexgyrationpropagation endurance, lowpower signal input through resonant excitation of vortex gyrations and extremely low energy dissipation in informationprocessing devices when using negligible damping materials.
Methods
Sample preparation
The five Py disk array is fabricated onto a 100nmthick siliconnitride membrane using electronbeam lithography and liftoff techniques. Each disk has a thickness of 60 nm and a diameter of 2 μm. The centertocenter distance between neighboring disks is 2.25 μm. An 800nmwide Cu stripline of 120 nm thickness (with a gold cap of 5 nm thickness) is deposited onto the first disk^{43}.
STXM measurement
Trajectories of the core motions of all five disks are directly observed using STXM by monitoring the outofplane core magnetizations at the MAXYMUS beamline (BESSY II; HelmholtzZentrum Berlin, Germany). The magnetic contrast is provided via XMCD at the Ni L_{3} absorption edge (around 852.7 eV). The measurements showing the core polatization^{48,49} are performed using negative circularpolarized xrays (where an upward/downward core appears as a dark/white spot), whereas the measurements showing the chirality configuration are performed with the sample tilted 60° with respect to the beam axis, using positive circularpolarized xrays (where a CW/CCW curling magnetization leads to a dark/bright contrast in the lower part of the disk). In the dynamic measurements, snapshot images of the individual core motions scanned in lateral steps of 8 nm are taken in time increments of 400 ps over a period of 60.8 ns after application of the field pulse at zero time.
Micromagnetic simulation
The LandauLifshitzGilbert (LLG) equation^{50,51} of motion of local magnetizations is numerically solved for the model geometry identical to that of the sample applied in the experimental measurement, using the OOMMF code^{46}. The material parameters corresponding to Py are as follows: saturation magnetization M_{s} = 780 × 10^{3} A/m and exchange stiffness constant A_{ex} = 1.3 × 10^{−11} J/m. For all the simulations, we used Gilbert damping constant α = 0.01.
Numerical calculation
Linearized coupled Thiele equations for vortex gyrations in N coupled disks are numerically solved by taking into account the potential energy as modified by the dipolar coupling between the NN disks. In the numerical calculation, we use the numerical values of , G = 1.77 × 10^{−12} Js/m^{2}, D = −4.66 × 10^{−14} Js/m^{2} and = 2.62 × 10^{−3} J/m^{2}, as obtained from the micromagnetic simulations performed on an isolated disk. The interaction strengths are determiend to be = 9.36 × 10^{−5} J/m^{2} and = 2.5 × 10^{−4} J/m^{2} according to the relation between Δω and the interaction strength coefficients^{36}, , where we obtained = 2π × 30 MHz and 2π × 15 MHz for p_{1}p_{2} = −1 and p_{1}p_{2} = 1, respectively, from further micromagnetic simulations on a coupled twodisk system of the same dimensions as those of the real sample.
Analytical derivation of dispersion for 1D infinite disk array
We obtained dispersion relations of 1D infinite disk arrays for different p and C orderings noted in the text, based on the Thiele equation of motion of a single vortex core in isolated disks, but by taking into account the potential energy modified by dipolar interaction between only the NN disks (see Supplementary Information C for further details.)
References
Kruglyak, V. V., Demokritov, S. O. & Grundler, J. Magnonics. J. Phys. D: Appl. Phys. 76, 264001 (2010).
Serga, A. A., Chumak, A. V. & Hillebrands, B. YIG magnonics. J. Phys. D: Appl. Phys. 43, 264002 (2010).
Kim, S. . K. Micromagnetic computer simulations of spin waves in nanometerscale patterned magnetic elements. J. Phys. D: Appl. Phys. 43, 264004 (2010).
Khitun, A., Nikonov, D. E., Bao, M., Galatsis, K. & Wang, K. L. Feasibility study of logic circuits with a spin wave bus. Nanotechnology 18, 465202 (2007).
Khitun, A., Bao, M. & Wang, K. L. Magnonic logic circuits. J. Phys. D: Appl. Phys. 43, 264005 (2010).
Choi, S., Lee, K.S., Guslienko, K. Y. & Kim, S.K. Strong Radiation of Spin Waves by Core Reversal of a Magnetic Vortex and their Wave Behaviors in Magnetic Nanowire Waveguides. Phys. Rev. Lett. 98, 087205 (2007).
Vasiliev, S. V., Kruglyak, V. V., Sokolovskii, M. L. & Kuchko, A. N. Spin wave interferometer employing a local nonuniformity of the effective magnetic field. J. Appl. Phys. 101, 113919 (2007).
Hertel, R., Wulfhekel, W. & Kirschner, J. DomainWall Induced Phase Shifts in Spin Waves. Phys. Rev. Lett. 93, 257202 (2004).
Schneider, T. et al. Realization of spinwave logic gates. Appl. Phys. Lett. 92, 022505 (2008).
Lee, K.S. & Kim, S.K. Conceptual design of spinwave logic gates based on a MachZehndertype spinwave interferometer for universal logic functions. J. Appl. Phys. 104, 053909 (2008).
Hansen, U., Demidov, V. & Demokritov, S. Dualfunction phase shifter for spinwave logic applications. Appl. Phys. Lett. 94, 252502 (2009).
Yu, Y., et al. Resonant amplification of vortexcore oscillations by coherent magneticfield pulses. Sci. Rep. 3, 1301 (2013).
Bayer, C. & Jorzick, J. [SpinWave Excitations in Finite Rectangular Elements] Spin Dynamics in Confined Magnetic Structures III, 1, (Hillebrands, B.& Thiaville, A. ed.) [57–100] (Springer Verlag, Berlin, 2006).
Cowburn, R. P. & Welland, M. E. Room Temperature Magnetic Quantum Cellular Automata. Science 287, 1466 (2000).
Suess, D., Schrefl, T., Fidler, J. & Tsiantos, V. Reversal dynamics of interacting circular nanomagnets. IEEE Trans. Mag. 37, 1960 (2001)
Lau, J. & Shaw, J. Magnetic nanostructures for advanced technologies: fabrication, metrology and challenges. J. Phys. D: Appl. Phys. 44, 303001 (2011).
Zhu, Y. [Magnetization dynamics using timeresolved magnetooptic microscopy.] Modern Techniques for Characterizing Magnetic Materials 1 (Zhu, Y. ed.) [517–575] (Springer, New York, 2005).
Chumak, A. V., Serga, A. A., Hillebrands, B. & Kostylev, M. P. Scattering of backward spin waves in a onedimensional magnonic crystal. Appl. Phys. Lett. 93, 022508 (2008).
Lee, K.S., Han, D.S. & Kim, S.K. Physical origin and generic control of magnonic band gaps of dipoleexchange spin waves in widthmodulatednanostrip waveguides. Phys. Rev. Lett. 102, 127202 (2009).
Kim, S.K., Lee, K.S. & Han, D.S. A gigahertzrange spinwave filter composed of widthmodulated nanostrip magnoniccrystal waveguides. Appl. Phys. Lett. 95, 082507 (2009).
Chumak, A. V. et al. Spinwave propagation in a microstructured magnonic crystal. Appl. Phys. Lett. 95, 262508 (2009).
Chumak, A. V. et al. Alllinear time reversal by a dynamic artificial crystal. Nat. Commun. 1, 141 (2010).
Puszkarski, H. & Krawczyk, M. Magnonic Crystals  the Magnetic Counterpart of Photonic Crystals. Solid State Phenom. 94, 125 (2003).
Kruglyak, V. V. et al. Imaging Collective Magnonic Modes in 2D Arrays of Magnetic Nanoelements. Phys. Rev. Lett. 104, 027201 (2010).
Tacchi, S. et al. Band Diagram of Spin Waves in a TwoDimensional Magnonic Crystal. Phys. Rev. Lett. 107, 127204 (2011).
Kobljanskyj, Y. et al. Nanostructured magnetic metamaterial with enhanced nonlinear properties. Sci. Rep. 2, 478 (2012).
Kostylev, M. P. et al. Propagating volume and localized spin wave modes on a lattice of circular magnetic antidots. J. Appl. Phys. 103, 07C507 (2008).
Neusser, S. & Grundler, D. Magnonics: Spin Waves on the Nanoscale. Adv. Mat. 21, 2927 (2009).
Neusser, S. et al. Anisotropic propagation and damping of spin waves in a nanopatterned antidot lattice. Phys. Rev. Lett. 105, 067208 (2010).
Chumak, A. V. et al. StorageRecovery Phenomenon in Magnonic Crystal. Phys. Rev. Lett. 108, 257207 (2012).
Ding, J., Kostylev, M. & Adeyeye, A. O. Magnonic crystal as a medium with tunable disorder on a periodical lattice. Phys. Rev. Lett. 107, 047205 (2011).
Shibata, J., Shigeto, K. & Otani Y. Dynamics of magnetostatically coupled vortices in magnetic nanodisks. Phys. Rev. B 67, 224404 (2003).
Shibata, J. & Otani, Y. Magnetic vortex dynamics in a twodimensional square lattice of ferromagnetic nanodisks. Phys. Rev. B 70, 012404 (2004).
Jung, H. et al. Observation of coupled vortex gyrations by 70pstime and 20nmspaceresolved fullfield magnetic transmission soft xray microscopy. Appl. Phys. Lett. 97, 222502 (2010).
Jung, H. et al. Tunable negligibleloss energy transfer between dipolarcoupled magnetic disks by stimulated vortex gyration. Sci. Rep. 1, 59 (2011).
Lee, K.S., Jung, H., Han, D.S. & Kim, S.K. Normal modes of coupled vortex gyration in to spatially separated magnetic nanodisks. J. Appl. Phys. 110, 113903 (2011).
Vogel, A., Drews, A., Kamionka, T., Bolte, M. & Meier, G. Influence of Dipolar Interaction on Vortex Dynamics in Arrays of Ferromagnetic Disks. Phys. Rev. Lett. 105, 037201 (2010).
Sugimoto, S. et al. Dynamics of coupled vortices in a pair of ferromagnetic disks. Phys. Rev. Lett. 106, 197203 (2011).
Vogel, A. et al. Coupled vortex oscillations in spatially separated Permalloy squares. Phys. Rev. Lett. 106, 137201 (2011).
Barman, S. Barman, A. & Otani, Y. Dynamics of 1D chains of magnetic vortices in response to local and global excitations. IEEE Trans. Magn. 46, 1342 (2010).
Barman, A., Barman, S., Kimura, T., Fukuma, Y. & Otani, Y. Gyration mode splitting in magnetostatically coupled magnetic vortices in an array. J. Phys. D: Appl. Phys. 43, 422001 (2010).
Vogel, A., Martens, M., Weigand, M. & Meier, G. Signal transfer in a chain of strayfield coupled ferromagnetic squares. Appl. Phys. Lett. 99, 042506 (2011).
Jung, H. et al. Logic Operations Based on MagneticVortexState Networks. ACS Nano 6, 3712 (2012).
Vogel, A., Drews, A., Weigand, M. & Meier, G. Direct imaging of phase relation in a pair of coupled vortex oscillators. AIP Adv. 2, 042180 (2012).
Jain, S. et al. From chaos to selective ordering of vortex cores in interacting mesomagnets. Nat. Commun. 3, 1330 (2012).
The object oriented micromagnetic framework (OOMMF) project at ITL/NIST, available at: http://math.nist.gov/oommf [Accessed November 18, 2007].
Thiele, A. A. SteadyState Motion of Magnetic Domains. Phys. Rev. Lett. 30, 230 (1973).
Chen, C. T., Sette, F., Ma, Y. & Modesti, S. Softxray magnetic circular dichroism at the L2,3 edges of nickel. Phys. Rev. B 42, 7262 (1990).
Kammerer, M. et al. Magnetic vortex core reversal by excitation of spin waves. Nat. Commun. 2, 279 (2011).
Landau, L. D. & Lifshitz, E. M. Theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8, 153 (1935).
Gilbert, T. L. A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Mag. 4, 3443 (2004).
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT & Future Planning (grant no. 2013003460). We acknowledge the support of Michael Bechtel, Eberhard Göring and BESSY II, HelmholtzZentrum Berlin. Financial support from the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich 668 and the Graduiertenkolleg 1286 is gratefully acknowledged. This work has been also supported by the excellence cluster ‘The Hamburg Centre for Ultrafast Imaging – Structure, Dynamics and Control of Matter at the Atomic Scale' of the Deutsche Forschungsgemeinschaft. P. F. acknowledges support from the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, U.S. Department of Energy (contract no. DEAC0205CH11231).
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S.K.K. and G.M independently led the project and conceived the main idea along with D.S.H. and A.V. respectively. D.S.H., K.S.L., H.J., P.F. and S.K.K. obtained preliminary experimental results of coupled vortex gyrations in different fivedisk array samples using magnetic transmission xray microscopy (MXTM) at the XM1 beamline (at Advanced Light Source, Lawrence Berkeley National Laboratory). A. V. prepared the samples discussed in the manuscript. A. V. and M. W. conducted the experimental measurements of vortex gyrations in the fivedisk array described in the manuscript using STXM at the MAXYMUS beamline (at BESSY II, HelmholtzZentrum Berlin). H.S. and G.S. contributed to the STXM setup and measurement. D.S.H. and S.K.K. derived the analytical equation of 1D infinite array and carried out the micromagnetic simulations and interpreted experimental and simulation results. S.K.K. and D.S.H. wrote the manuscript. G.M., A.V. and P. F discussed the results and commented on the manuscript.
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Han, DS., Vogel, A., Jung, H. et al. Wave modes of collective vortex gyration in dipolarcoupleddotarray magnonic crystals. Sci Rep 3, 2262 (2013). https://doi.org/10.1038/srep02262
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